Directed Acyclic Graph networks to characterize phase space evolution with application to musical composition and industrial maintenance

Phase space networks built on proximity of reconstructed state space points from time series are useful for system identification and characterization. We consider causal proximity networks where directed edges connect vertices corresponding to future states to past states. Here, we introduce a new network which captures how induced proximity network subgraphs evolve in time. Due to the causal nature of the proximity network these subgraphs are directed acyclic graphs (DAG) and the new network is a transition network. This network can be summarized using entropy-like properties. We show how to incorporate this new network into a data analysis pipeline and highlight its potential to detect dynamical change and the evolving geometry of state space. Applications to music composition and industrial maintenance are used to illustrate the promise of the approach.